I just finished a course in Calculus 2, and I was surprised that there are many theorems that consider pre-calculus theorems like "theorems of polynomials" such as the Fundamental Theorem of Algebra, Descartes' Rule of Signs, the Factor Theorem,….. . but We didn't study these theorems in high school or college. However, after finishing Calculus 2, pre-calculus books seem too easy for me. Reading one seems like a waste of time. Although some of the the theorems are very basic that I can learn them just from youtube, like the Factor Theorem, I am pretty sure that there is much in the field of basic algebra that covers only polynomials and is not elementary. It considers pretty advanced topics like not every polynomial being solvable. So my question is: Is there a book that covers "theorems of polynomials" with much depth, detail, and rigor and what course cover these topics?
Books on polynomial algebra with much detail than pre-calc books
algebra-precalculusbook-recommendationpolynomials
Related Solutions
Honestly, you don't need a huge algebra background. Also, In algebraic/geometric topology one does not need a huge point set topology. I think you've enough point set topology background. Basic notions of groups such as groups, subgroups, and homomorphism/isomorphism are needed pretty much all the time. You should be really comfortable with free abelian groups those are the main objects(Homology and homotopy groups) in algebraic topology. When you'll compute fundamental groups, you will find that there are spaces where fundamental groups cannot be easily written explicitly, for example, Kleine bottle. So, you should be comfortable with generators and relations. To compute some homology/cohomology groups sometimes, you will use the tensor product, Free product(many many times), $Hom(A, B)$, $Tor(A, B)$ and $Ext(A, B)$. You can use them as a black box, but understanding them clearly will be fun for sure. If you're familiar with exact sequences, and basic notions of modules that will be extremely helpful. Fun fact: you'll use the first isomorphism theorem many times. I hope this helps.
For control theory, given your background, you should start with
General (point-set) topology. You can find textbook recommendations here. McCleary's book is the fastest, Morris' book is the slowest.
Then differential geometry/Riemannian geometry (it will serve you better than a differential topology class): Make sure to take a more advanced Differential Geometry class, not "Curves and surfaces:" The advanced class should cover differentiable manifolds, connections, Riemannian metrics.
My favorite for Riemannian Geometry is do Carmo's "Riemannian Geometry," mostly chapters 0 through 4. It is also the fastest.
Another good option is
Abraham, Ralph; Marsden, Jerrold E., Foundations of mechanics. 2nd ed., rev., enl., and reset. With the assistance of Tudor Ratiu and Richard Cushman, Reading, Massachusetts: The Benjamin/Cummings Publishing Company, Inc., Advanced Book Program. m-XVI, XXII, 806 p. $ 36.50 (1978). ZBL0393.70001.
They will cover differentiable manifolds, forms, Frobenius theorem, basics of Riemannian geometry...
Frequently (but not always) these are covered in a differential topology class (Guillemin and Pollack while a really good book, will not help you here). This will clear most of the language problems you are currently facing. But, critically, you need an advisor to point you in the right direction since there will be no general-purpose courses helping with control theory beyond that point.
Edit. What you really need for CT is "sub-Finslerian differential geometry," which is a combination of the theory of general (typically non-integrable) distributions and Finsler metrics defined on such distributions. A Riemannian Geometry class will teach you about Riemannian metrics. The formalism of Finsler metrics is similar, but the technicalities are much harder. For sub-Riemannian geometries, take a look here:
Montgomery, Richard, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs 91. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-1391-9/hbk). xx, 259 p. (2002). ZBL1044.53022.
For Finsler geometry, the standard reference is
Bao, D.; Chern, S.-S.; Shen, Z., An introduction to Riemann-Finsler geometry, Graduate Texts in Mathematics. 200. New York, NY: Springer. xx, 431 p. (2000). ZBL0954.53001.
For sub-Finslerian geometry, you have to read research papers, there are no textbook treatments.
Best Answer
I think the topic you're looking for is "theory of equations." It's sort of old-fashioned, and has been absorbed by algebra, but there are still texts and websites that treat the subject. Here's the wiki page:
https://en.wikipedia.org/wiki/Theory_of_equations
and here's a link to a book that's 120 years old, but probably has the sort of stuff you're looking for:
https://www.amazon.com/Introduction-Algebraic-Equations-Leonard-Dickson/dp/B00AOX1S7C/ref=sr_1_6
As others are suggesting, you could take the algebraic route (and you should, eventually) but "theory of equations" seems more direct to your question.